Optimal. Leaf size=394 \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} f+5 \sqrt{b} d\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}-\frac{2 a^2 f x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a^{9/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )}{24 b}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left (9 d+7 f x^2\right )+\frac{2 a d x \sqrt{a+b x^4}}{21 b}-\frac{a e x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b} \]
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Rubi [A] time = 0.334251, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367, Rules used = {1833, 1252, 780, 195, 217, 206, 1274, 1280, 1198, 220, 1196} \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \left (7 \sqrt{a} f+5 \sqrt{b} d\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}-\frac{a^2 e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}-\frac{2 a^2 f x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{2 a^{9/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )}{24 b}+\frac{1}{63} x^5 \sqrt{a+b x^4} \left (9 d+7 f x^2\right )+\frac{2 a d x \sqrt{a+b x^4}}{21 b}-\frac{a e x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b} \]
Antiderivative was successfully verified.
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Rule 1833
Rule 1252
Rule 780
Rule 195
Rule 217
Rule 206
Rule 1274
Rule 1280
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int x^3 \left (c+d x+e x^2+f x^3\right ) \sqrt{a+b x^4} \, dx &=\int \left (x^3 \left (c+e x^2\right ) \sqrt{a+b x^4}+x^4 \left (d+f x^2\right ) \sqrt{a+b x^4}\right ) \, dx\\ &=\int x^3 \left (c+e x^2\right ) \sqrt{a+b x^4} \, dx+\int x^4 \left (d+f x^2\right ) \sqrt{a+b x^4} \, dx\\ &=\frac{1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt{a+b x^4}+\frac{1}{2} \operatorname{Subst}\left (\int x (c+e x) \sqrt{a+b x^2} \, dx,x,x^2\right )+\frac{1}{63} (2 a) \int \frac{x^4 \left (9 d+7 f x^2\right )}{\sqrt{a+b x^4}} \, dx\\ &=\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b}+\frac{1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt{a+b x^4}+\frac{\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac{(2 a) \int \frac{x^2 \left (21 a f-45 b d x^2\right )}{\sqrt{a+b x^4}} \, dx}{315 b}-\frac{(a e) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^2\right )}{8 b}\\ &=\frac{2 a d x \sqrt{a+b x^4}}{21 b}-\frac{a e x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b}+\frac{1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt{a+b x^4}+\frac{\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}+\frac{(2 a) \int \frac{-45 a b d-63 a b f x^2}{\sqrt{a+b x^4}} \, dx}{945 b^2}-\frac{\left (a^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^2\right )}{16 b}\\ &=\frac{2 a d x \sqrt{a+b x^4}}{21 b}-\frac{a e x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b}+\frac{1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt{a+b x^4}+\frac{\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac{\left (a^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^2}{\sqrt{a+b x^4}}\right )}{16 b}+\frac{\left (2 a^{5/2} f\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 b^{3/2}}-\frac{\left (2 a^2 \left (5 \sqrt{b} d+7 \sqrt{a} f\right )\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{105 b^{3/2}}\\ &=\frac{2 a d x \sqrt{a+b x^4}}{21 b}-\frac{a e x^2 \sqrt{a+b x^4}}{16 b}+\frac{2 a f x^3 \sqrt{a+b x^4}}{45 b}-\frac{2 a^2 f x \sqrt{a+b x^4}}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{63} x^5 \left (9 d+7 f x^2\right ) \sqrt{a+b x^4}+\frac{\left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac{a^2 e \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{16 b^{3/2}}+\frac{2 a^{9/4} f \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^4}}-\frac{a^{7/4} \left (5 \sqrt{b} d+7 \sqrt{a} f\right ) \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 b^{7/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.630121, size = 215, normalized size = 0.55 \[ \frac{\sqrt{a+b x^4} \left (63 e \left (\sqrt{b} x^2 \left (a+2 b x^4\right )-\frac{a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{\sqrt{\frac{b x^4}{a}+1}}\right )+168 \sqrt{b} c \left (a+b x^4\right )-\frac{144 a \sqrt{b} d x \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+144 \sqrt{b} d x \left (a+b x^4\right )-\frac{112 a \sqrt{b} f x^3 \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{\sqrt{\frac{b x^4}{a}+1}}+112 \sqrt{b} f x^3 \left (a+b x^4\right )\right )}{1008 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 380, normalized size = 1. \begin{align*}{\frac{f{x}^{7}}{9}\sqrt{b{x}^{4}+a}}+{\frac{2\,{x}^{3}af}{45\,b}\sqrt{b{x}^{4}+a}}-{{\frac{2\,i}{15}}f{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{{\frac{2\,i}{15}}f{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{e{x}^{2}}{8\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{ae{x}^{2}}{16\,b}\sqrt{b{x}^{4}+a}}-{\frac{{a}^{2}e}{16}\ln \left ({x}^{2}\sqrt{b}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{d{x}^{5}}{7}\sqrt{b{x}^{4}+a}}+{\frac{2\,adx}{21\,b}\sqrt{b{x}^{4}+a}}-{\frac{2\,{a}^{2}d}{21\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{c}{6\,b} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}} c}{6 \, b} + \int{\left (f x^{6} + e x^{5} + d x^{4}\right )} \sqrt{b x^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )} \sqrt{b x^{4} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.29747, size = 212, normalized size = 0.54 \begin{align*} \frac{a^{\frac{3}{2}} e x^{2}}{16 b \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} d x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{3 \sqrt{a} e x^{6}}{16 \sqrt{1 + \frac{b x^{4}}{a}}} + \frac{\sqrt{a} f x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} - \frac{a^{2} e \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + c \left (\begin{cases} \frac{\sqrt{a} x^{4}}{4} & \text{for}\: b = 0 \\\frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b} & \text{otherwise} \end{cases}\right ) + \frac{b e x^{10}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{4}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{4} + a}{\left (f x^{3} + e x^{2} + d x + c\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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